Symmetry-protected third-order exceptional points in staggered flatband rhombic lattices

Yingying Zhang; Shiqiang Xia; Xingdong Zhao; Lu Qin; Xuejing Feng; Wenrong Qi; Yajing Jiang; Hai Lu; Daohong Song; Liqin Tang; Zunlue Zhu; Wuming Liu; Yufang Liu

doi:10.1364/PRJ.478167

[1] L. Feng, R. El-Ganainy, L. Ge. Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics, 11, 752-762(2017).

[2] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, D. N. Christodoulides. Non-Hermitian physics and PT symmetry. Nat. Phys., 14, 11-19(2018).

[3] Ş. K. Özdemir, S. Rotter, F. Nori, L. Yang. Parity–time symmetry and exceptional points in photonics. Nat. Mater., 18, 783-798(2019).

[4] M.-A. Miri, A. Alu. Exceptional points in optics and photonics. Science, 363, eaar7709(2019).

[5] M. Parto, Y. G. Liu, B. Bahari, M. Khajavikhan, D. N. Christodoulides. Non-Hermitian and topological photonics: optics at an exceptional point. Nanophotonics, 10, 403-423(2021).

[6] E. J. Bergholtz, J. C. Budich, F. K. Kunst. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys., 93, 015005(2021).

[7] L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, P. Xue. Observation of non-Bloch parity-time symmetry and exceptional points. Phys. Rev. Lett., 126, 230402(2021).

[8] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett., 100, 103904(2008).

[9] S. Klaiman, U. Günther, N. Moiseyev. Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett., 101, 080402(2008).

[10] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, M. Khajavikhan. Parity-time–symmetric microring lasers. Science, 346, 975-978(2014).

[11] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, X. Zhang. Single-mode laser by parity-time symmetry breaking. Science, 346, 972-975(2014).

[12] S. Malzard, C. Poli, H. Schomerus. Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry. Phys. Rev. Lett., 115, 200402(2015).

[13] H. Shen, B. Zhen, L. Fu. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett., 120, 146402(2018).

[14] S. Xia, D. Kaltsas, D. Song, I. Komis, J. Xu, A. Szameit, H. Buljan, K. G. Makris, Z. Chen. Nonlinear tuning of PT symmetry and non-Hermitian topological states. Science, 372, 72-76(2021).

[15] W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, L. Yang. Exceptional points enhance sensing in an optical microcavity. Nature, 548, 192-196(2017).

[16] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, M. Khajavikhan. Enhanced sensitivity at higher-order exceptional points. Nature, 548, 187-191(2017).

[17] M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, M. Khajavikhan. Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity. Nature, 576, 70-74(2019).

[18] R. Duggan, S. A. Mann, A. Alù. Limitations of sensing at an exceptional point. ACS Photon., 9, 1554-1566(2022).

[19] S. Longhi. PT symmetry and antisymmetry by anti-Hermitian wave coupling and nonlinear optical interactions. Opt. Lett., 43, 4025-4028(2018).

[20] I. Mandal, E. J. Bergholtz. Symmetry and higher-order exceptional points. Phys. Rev. Lett., 127, 186601(2021).

[21] P. Delplace, T. Yoshida, Y. Hatsugai. Symmetry-protected multifold exceptional points and their topological characterization. Phys. Rev. Lett., 127, 186602(2021).

[22] K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, M. Ueda. Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics. Nat. Commun., 10, 297(2019).

[23] K. Kawabata, K. Shiozaki, M. Ueda, M. Sato. Symmetry and topology in non-Hermitian physics. Phys. Rev. X, 9, 041015(2019).

[24] T. Yoshida, T. Mizoguchi, Y. Kuno, Y. Hatsugai. Square-root topological phase with time-reversal and particle-hole symmetry. Phys. Rev. B, 103, 235130(2021).

[25] J. D. H. Rivero, L. Ge. Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra. Phys. Rev. B, 103, 014111(2021).

[26] K. Sun, Z. Gu, H. Katsura, S. D. Sarma. Nearly flatbands with nontrivial topology. Phys. Rev. Lett., 106, 236803(2011).

[27] N. B. Kopnin, T. T. Heikkilä, G. E. Volovik. High-temperature surface superconductivity in topological flat-band systems. Phys. Rev. B, 83, 220503(2011).

[28] L. Balents, C. R. Dean, D. K. Efetov, A. F. Young. Superconductivity and strong correlations in moiré flat bands. Nat. Phys., 16, 725-733(2020).

[29] P. Wang, Y. Zheng, X. Chen, C. Huang, Y. V. Kartashov, L. Torner, V. V. Konotop, F. Ye. Localization and delocalization of light in photonic Moiré lattices. Nature, 577, 42-46(2020).

[30] R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Meja-Cortés, S. Weimann, A. Szameit, M. I. Molina. Observation of localized states in Lieb photonic lattices. Phys. Rev. Lett., 114, 245503(2015).

[31] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, R. R. Thomson. Observation of a localized flat-band state in a photonic Lieb lattice. Phys. Rev. Lett., 114, 245504(2015).

[32] C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, D. N. Krizhanovskii. Exciton polaritons in a two-dimensional Lieb lattice with spin-orbit coupling. Phys. Rev. Lett., 120, 097401(2018).

[33] D. Leykam, S. Flach. Perspective: photonic flatbands. APL Photon., 3, 070901(2018).

[34] L. Tang, D. Song, S. Xia, S. Xia, J. Ma, W. Yan, Y. Hu, J. Xu, D. Leykam, Z. Chen. Photonic flat-band lattices and unconventional light localization. Nanophotonics, 9, 1161-1176(2020).

[35] R. A. V. Poblete. Photonic flat band dynamics. Adv. Phys.: X, 6, 1878057(2021).

[36] J.-W. Rhim, B.-J. Yang. Classification of flat bands according to the band-crossing singularity of Bloch wave functions. Phys. Rev. B, 99, 045107(2019).

[37] J. Ma, J.-W. Rhim, L. Tang, S. Xia, H. Wang, X. Zheng, S. Xia, D. Song, Y. Hu, Y. Li, B.-J. Yang, D. Leykam, Z. Chen. Direct observation of flatband loop states arising from nontrivial real-space topology. Phys. Rev. Lett., 124, 183901(2020).

[38] S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, A. Szameit. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater., 16, 433-438(2017).

[39] T. Biesenthal, M. Kremer, M. Heinrich, A. Szameit. Experimental realization of PT symmetric flat bands. Phys. Rev. Lett., 123, 183601(2019).

[40] L. Jin. Flat band induced by the interplay of synthetic magnetic flux and non-Hermiticity. Phys. Rev. A, 99, 033810(2019).

[41] B. Qi, L. Zhang, L. Ge. Defect states emerging from a non-Hermitian flatband of photonic zero modes. Phys. Rev. Lett., 120, 093901(2018).

[42] S. Longhi, L. Feng. Invited article: mitigation of dynamical instabilities in laser arrays via non-Hermitian coupling. APL Photon., 3, 060802(2018).

[43] D. Leykam, S. Flach, Y. D. Chong. Flat bands in lattices with non-Hermitian coupling. Phys. Rev. B, 96, 064305(2017).

[44] S. Xia, C. Danieli, Y. Zhang, X. Zhao, H. Lu, L. Tang, D. Li, D. Song, Z. Chen. Higher-order exceptional point and Landau–Zener Bloch oscillations in driven non-Hermitian photonic Lieb lattices. APL Photon., 6, 126106(2021).

[45] S. Xia, A. Ramachandran, S. Xia, D. Li, X. Liu, L. Tang, Y. Hu, D. Song, J. Xu, D. Leykam, S. Flach, Z. Chen. Unconventional flatband line states in photonic Lieb lattices. Phys. Rev. Lett., 121, 263902(2018).

[46] J.-W. Rhim, K. Kim, B.-J. Yang. Quantum distance and anomalous landau levels of flat bands. Nature, 584, 59-63(2020).

[47] L. Ge. Non-Hermitian lattices with a flat band and polynomial power increase. Photon. Res., 6, A10-A17(2018).

[48] Y.-X. Xiao, K. Ding, R.-Y. Zhang, Z. H. Hang, C. T. Chan. Exceptional points make an astroid in non-Hermitian Lieb lattice: evolution and topological protection. Phys. Rev. B, 102, 245144(2020).

[49] R. D. J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, Y. N. Joglekar. Observation of slowly decaying eigenmodes without exceptional points in Floquet dissipative synthetic circuits. Commun. Phys., 1, 88(2018).

[50] M. A. Quiroz-Juárez, A. Perez-Leija, K. Tschernig, B. M. Rodríguez-Lara, O. S. Magana-Loaiza, K. Busch, Y. N. Joglekar, R. de J. León-Montiel. Exceptional points of any order in a single, lossy waveguide beam splitter by photon-number-resolved detection. Photon. Res., 7, 862-867(2019).

[51] A. Ramachandran, A. Andreanov, S. Flach. Chiral flat bands: existence, engineering, and stability. Phys. Rev. B, 96, 161104(2017).

[52] A. R. Kolovsky, A. Ramachandran, S. Flach. Topological flat Wannier-Stark bands. Phys. Rev. B, 97, 045120(2018).

[53] S. Xia, C. Danieli, W. Yan, D. Li, S. Xia, J. Ma, H. Lu, D. Song, L. Tang, S. Flach, Z. Chen. Observation of quincunx-shaped and dipole-like flatband states in photonic rhombic lattices without band-touching. APL Photon., 5, 016107(2020).

[54] M. Kremer, I. Petrides, E. Meyer, M. Heinrich, O. Zilberberg, A. Szameit. A square-root topological insulator with non-quantized indices realized with photonic Aharonov-Bohm cages. Nat. Commun., 11, 907(2020).

[55] S. M. Zhang, L. Jin. Localization in non-Hermitian asymmetric rhombic lattice. Phys. Rev. Res., 2, 033127(2020).

[56] S. Xia, Y. Zhang, Z. Li, L. Qin, C. Yang, H. Lu, J. Zhang, X. Zhao, Z. Zhu. Band evolution and Landau-Zener Bloch oscillations in strained photonic rhombic lattices. Opt. Express, 29, 37503-37514(2021).

[57] S. Mukherjee, M. Di Liberto, P. Öhberg, R. R. Thomson, N. Goldman. Experimental observation of Aharonov-Bohm cages in photonic lattices. Phys. Rev. Lett., 121, 075502(2018).

[58] S. Mukherjee, R. R. Thomson. Observation of localized flat-band modes in a quasi-one-dimensional photonic rhombic lattice. Opt. Lett., 40, 5443-5446(2015).

[59] B. Midya, H. Zhao, L. Feng. Non-Hermitian photonics promises exceptional topology of light. Nat. Commun., 9, 2674(2018).

[60] S. Ke, D. Zhao, J. Liu, Q. Liu, Q. Liao, B. Wang, P. Lu. Topological bound modes in anti-PT-symmetric optical waveguide arrays. Opt. Express, 27, 13858-13870(2019).

[61] L. Du, Y. Zhang, J.-H. Wu. Controllable unidirectional transport and light trapping using a one-dimensional lattice with non-Hermitian coupling. Sci. Rep., 10, 1(2020).

[62] R. Keil, C. Poli, M. Heinrich, J. Arkinstall, G. Weihs, H. Schomerus, A. Szameit. Universal sign control of coupling in tight-binding lattices. Phys. Rev. Lett., 116, 213901(2016).

[63] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani. PT-symmetric optical lattices. Phys. Rev. A, 81, 063807(2010).

[64] J. Schnabel, H. Cartarius, J. Main, G. Wunner, W. D. Heiss. PT-symmetric waveguide system with evidence of a third-order exceptional point. Phys. Rev. A, 95, 053868(2017).

[65] S. Longhi, D. Gatti, G. D. Valle. Robust light transport in non-Hermitian photonic lattices. Sci. Rep., 5, 13376(2015).

[66] S. Longhi. Phase transitions in a non-Hermitian Aubry-André-Harper model. Phys. Rev. B, 103, 054203(2021).

[67] K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, C. T. Chan. Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization. Phys. Rev. X, 6, 021007(2016).

[68] M. Ezawa. Nonlinear non-Hermitian higher-order topological laser. Phys. Rev. Res., 4, 013195(2022).

[69] G. G. Pyrialakos, H. Ren, P. S. Jung, M. Khajavikhan, D. N. Christodoulides. Thermalization dynamics of nonlinear non-Hermitian optical lattices. Phys. Rev. Lett., 128, 213901(2022).

[70] I. Komis, D. Kaltsas, S. Xia, H. Buljan, Z. Chen, K. Makris. Robustness versus sensitivity in non-Hermitian topological lattices probed by pseudospectra. arXiv(2022).